By the sound of things you have a mod pack installed. Those often have more than just TerraFirmaCraft installed. So you can either modify the mod pack, or use one that only has TFC, or install TFC following the instructions here:http://wiki.terrafirmacraft.com/Installation

Alternatively you could (and shoudl!) use MultiMC since the default Minecraft launcher isn't very good when it comes to managing mods.
Download MultiMC here: https://multimc.org/
Extract the MultiMC .zip file to a location of your choosing. For example C:\games\MultiMC
Run MultiMC. The first time MultiMC starts it will ask you where Java is installed. Usually selecting path "java" will suffice. Just make sure you have 64-bit Java installed. If not, uninstall Java and download the latest 64-bit version here: https://www.java.com/en/download/manual.jsp

Now add a new instance. Call it HappyDiggers TFC (since you're making it to join our server)
Make sure you select Minecraft version 1.7.10

MultiMC will tell you it needs a Minecraft account in order to download Minecraft assets:
We'll add an account below. First let's finish the instance

Select the HappyDiggers TFC instance and click "edit instance":

Click "Install Forge":
The current version of TerraFirmaCraft was build for Forge version 10.13.4.1558 so select that version.

Select the HappyDiggers TFC instance and now click "Instance Folder". This will open the folder where all the files for this instance are installed:

Now go back to MultiMC.
Select the HappyDiggers TFC instance and click "play".
MultiMC will tell you (again) that you need to add a Minecraft account and ask you to add one. Click "Yes":

Click "Add":
Now add your Minecraft account credentials:

Once your account has been added, close the account settings panel.
Select the HappyDiggers TFC instance again, and click "Play" again.
MultiMC will download Minecraft, Forge and a bunch of libraries.
Minecraft should now start with the latest TerraFirmaCraft mod installed.
If you want to add more mods you can do so by adding them to the mods folder like we dit with TerraFirmaCraft above. Just be sure to check which mods are whitelisted on our server first.

One note: currently the Minecraft HTTP servers that serve the assets seems to be down. http://xpaw.ru/mcstatus/
They're down for about 4 days now. Because of this mod launchers won't be able to download assets from Mojang. This issue usually resolves itself when the sys admin comes back from the weekend and pushes the reset button on the server